Solve (x^2-2)^2-(x^2+2)^2

$$( x ^ { 2 } - 2 ) ^ { 2 } - ( x ^ { 2 } + 2 ) ^ { 2 }$$

Evaluate

$$-8x^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-2\right)^{2}.

$$\left(x^{2}\right)^{2}-4x^{2}+4-\left(x^{2}+2\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-4x^{2}+4-\left(x^{2}+2\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+2\right)^{2}.

$$x^{4}-4x^{2}+4-\left(\left(x^{2}\right)^{2}+4x^{2}+4\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-4x^{2}+4-\left(x^{4}+4x^{2}+4\right)$$

To find the opposite of x^{4}+4x^{2}+4, find the opposite of each term.

$$x^{4}-4x^{2}+4-x^{4}-4x^{2}-4$$

Combine x^{4} and -x^{4} to get 0.

$$-4x^{2}+4-4x^{2}-4$$

Combine -4x^{2} and -4x^{2} to get -8x^{2}.

$$-8x^{2}+4-4$$

Subtract 4 from 4 to get 0.

$$-8x^{2}$$

Expand

$$-8x^{2}$$

Solution Steps

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(x^{2}-2\right)^{2}.

$$\left(x^{2}\right)^{2}-4x^{2}+4-\left(x^{2}+2\right)^{2}$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-4x^{2}+4-\left(x^{2}+2\right)^{2}$$

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x^{2}+2\right)^{2}.

$$x^{4}-4x^{2}+4-\left(\left(x^{2}\right)^{2}+4x^{2}+4\right)$$

To raise a power to another power, multiply the exponents. Multiply 2 and 2 to get 4.

$$x^{4}-4x^{2}+4-\left(x^{4}+4x^{2}+4\right)$$

To find the opposite of x^{4}+4x^{2}+4, find the opposite of each term.

$$x^{4}-4x^{2}+4-x^{4}-4x^{2}-4$$

Combine x^{4} and -x^{4} to get 0.

$$-4x^{2}+4-4x^{2}-4$$

Combine -4x^{2} and -4x^{2} to get -8x^{2}.

$$-8x^{2}+4-4$$

Subtract 4 from 4 to get 0.

$$-8x^{2}$$